Models with limited dependent variables
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Models with limited dependent variables. Doctoral Program 2006-2007 Katia Campo. 3. Nested Logit Model. (K.Train, Ch.4, Franses and Paap Ch.5). 3. Nested Logit Model. Choice between J>2 alternatives which can be grouped into subsets based on differences in substitution pattern Example:.

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Models with limited dependent variables

Models with limited dependent variables

Doctoral Program 2006-2007

Katia Campo


3 nested logit model

3. Nested Logit Model

(K.Train, Ch.4, Franses and Paap Ch.5)


3 nested logit model1

3. Nested Logit Model

  • Choice between J>2 alternatives which can be grouped into subsets based on differences in substitution pattern

  • Example:

Choice of transportation mode

Car

Transit

Carpool

Bus

Train

Car alone


3 nested logit model2

3. Nested Logit Model

  • Assumption cumulative distribution of error terms

  • Probability


3 nested logit model3

3. Nested Logit Model

  • Probability can be decomposed into 2 logit models

(1)

(2)


3 nested logit model4

3. Nested Logit Model

  • Link between Pni | Bk * Pn Bk (upper and lower model level) through inclusive value Ink

  • To comply with RUM k must be in the [0,1] interval

  • IIA within, not across nests


3 nested logit model5

3. Nested Logit Model

Estimation

  • Joint estimation

  • Sequential estimation

    • Estimate lower model

    • Compute inclusive value

    • Estimate upper model with inclusive value as explanatory variable

      Disadvantages sequential estimation

    • Add fourth step, using the parameter estimates as starting values for joint estimation


3 nested logit model6

3. Nested Logit Model

  • Example: Location choices by French firms in Eastern and Western Europe

  • Dependent var.: probability of choosing location j Pj=P(πj> πk  k≠j )

  • Location choices are likely to have a nested structure (non-IIA)

    • First, select region (East or Western Europe)

    • Next, select country within region

Disdier and Mayer (2004)


3 nested logit model7

3. Nested Logit Model

  • Example 1: Location choices by French firms in Eastern and Western Europe

Location choice

E.Eur

W.Eur

………

………

CN

C1

CJ+1

CJ

Disdier and Mayer (2004)


3 nested logit model8

3. Nested Logit Model

  • Location choices: Data

    • 1843 location decisions in Europe from 1980 to 1999 (official statistics)

    • 19 host countries (13 W.Eur, 6 E.Eur)

Disdier and Mayer (2004)


3 nested logit model9

3. Nested Logit Model

  • Location choices: Data

    • NFFrench firms already located in the country

    • GDPGDP

    • GPP/CAPGDP per capita

    • DISTDistance France – host country

    • WAverage wage per capita (manufacturing)

    • UNEMPLunemployment rate

    • EXCHRExchange rate volatility

    • FREEFree country

    • PNFREEPartly free and not free country

    • PR1Country with political rights rated 1

    • PR2Country with political rights rated 2

    • PR345Country with political rights rated 3,4,5

    • PR67Country with political rights rated 6,7

    • LIAnnual liberalization index

    • CLICumulative liberalization index

    • ASSOC=1 if an association agreement is signed

Disdier and Mayer (2004)


Models with limited dependent variables

3. Nested Logit Model

Disdier and Mayer (2004)


Models with limited dependent variables

3. Nested Logit Model

Disdier and Mayer (2004)


Models with limited dependent variables

3. Nested Logit Model

  • Example 2: Shopping centre choice


Models with limited dependent variables

3. Nested Logit Model

  • Example 2: Shopping centre choice


Models with limited dependent variables

3. Nested Logit Model

  • Example 2: Shopping centre choice


3 nested logit model10

3. Nested Logit Model

  • (Purchase) incidence and choice models can be linked in the same way

Include incl.value 

Purchase decision

No purchase

Purchase

...

(See above: Bucklin & Gupta)

Altern. J

Altern.1


3 nested logit model11

3. Nested Logit Model

  • Other examples

    • Private label versus nationals brands

    • Same versus other brand

    • Fixed rate (low risk) versus variable rate (high risk) investments

    • Choice of transportation mode

    • ....


4 probit model

4. Probit Model

(K.Train, Ch.5 ; Franses and Paap, Ch.4-5)


4 probit model1

4. Probit Model

  • Based on the general RUM-model

  • Ass.: error terms are distributed normal with a mean vector of zero and covariance matrix 

  • density function:


4 probit model2

4. Probit Model

Logit or Probit?

  • Trade-off between tractability and flexibility

    • Closed-form expression of the integral for Logit, not for Probit models

    • Probit allows for random taste variation, can capture any substitution pattern, allows for correlated error terms and unequal error variances

 Dependent on the specifics of the choice situation


4 probit model3

4. Probit Model

Estimation

  • Approximation of the multidimensional integral

    • Non-simulation procedures (see Kamakura 1989)

      Can usually only be applied to restricted cases and/or provide inaccurate estimations

    • Simulation procedures (see Geweke et al.1994, Train)


4 probit model4

4. Probit Model

Simulation-based estimation(binary probit, CFS)

  • Step 1

    • For each observation n=1, ..., N draw r ~ N(0,1), (r = 1, ......., R: repetitions)

    • Initialize y_count= 0, =mt (starting values)

    • Compute y*rn = xn mt + L r ; L= choleski factor (LL’= )

    • Evaluate: y*rn >0  y_count= y_count+1

    • Repeat R times

Weeks (1997)


4 probit model5

4. Probit Model

Simulation-based estimation(binary probit, CFS)

  • Step 2: calculate probabilities

    Pn| mt= y_count/R

  • Step 3: Form the simulated LL function

    SLL=  n yn ln(Pn|mt)+(1-yn) ln(1-Pn|mt)

  • Step 4: Check convergence criteria (SLL(mt)- SLL(mt-1))

  • Step 5: Update mt: mt+1 = mt + v

  • Step 6: Iterate (until convergence)

Weeks (1997)


4 probit model6

4. Probit Model

Simulation-based estimation: MNProbit

  • Based on same principles

  • More efficient simulation procedures (see Train)

  • Identification: normalization of level and scale

    • Re-express model in utility differences

    • Normalization of varcov matrix (see Train)


4 probit model7

4. Probit Model

  • Random taste variation

  • Model with random coefficients

    • E.g.:n~N(b,W)

    • Unj = b’xnj+ *’n xnj + nj

    • = b’xnj+ nj

    • nj : correlated error terms (dep.on xnj, see Train)


4 probit model8

4. Probit Model

  • Substitution patterns

  • Full covariance matrix (no parameter restrictions) unrestricted substitution patterns

  • Structured covariance matrix (restrictions on some covariance parameters)

  •  the structure imposed on  determines the substitution pattern and may allow to reduce the number of parameters to be estimated


4 probit model9

4. Probit Model

  • Example (Kamakura and Srivastava 1984): random utility components ni, nj are more (less) highly correlated when i and j are more (less) similar on important attributes

(dij = weighted eucledian distance between i & j)


4 probit model10

4. Probit Model

  • Examples

    • Choice models at brand-size level: correlation between ≠ sizes of same brand (Chintagunta 1992)

MNL model

gives biased

estimates

of price

elasticity


4 probit model11

4. Probit Model

  • Examples

    • Firm innovation (Harris et al. 2003)

      • Binary probit model for innovative status (innovation occurred or not)

      • Based on panel data  correlation of innovative status over time: unobserved heterogeneity related to management ability and/or strategy


Models with limited dependent variables

Model (2)-(4) account for unobserved heterogeneity (ρ) -> superior results


4 probit model12

4. Probit Model

  • Examples

    • Dynamics of individual health (Contoyannis, Jones and Nigel 2004)

      • Binary probit model for health status (healthy or not)

      • Survey data for several years  correlation over time (state dependence) + individual-specific (time-invariant) random coefficient


4 probit model13

4. Probit Model

  • Examples

    • Choice of transportation mode (Linardakis and Dellaportas 2003)  Non-IIA substitution patterns


Logit vs probit

Logit vs Probit


5 ordered logit model

5. Ordered Logit Model


5 ordered logit model1

5. Ordered Logit Model

  • Choice between J>2 ordered ‘alternatives’

  • Ordinal dependent variable y = 1, 2, ... J, with

    rank(1) < rank(2) < ... < rank(J)

  • Example:

    • Purchase of 1, 2, ... J units

    • Evaluation on a J-point scale ranging from, e.g., ‘dislike very much’ to ‘like very much’


5 ordered logit model2

5. Ordered Logit Model

  • Suppose yi* is a continous latent variable which

    • is a linear function of the explanatory variables

      yn* = Xn + n

    • and can be ‘mapped’ on an ordered multinomial variable as follows:

      yn= 1 if 0 < yn*  1

      yn= j if j-1 < yn*  j

      yn= J if J-1 < yn*  J

      0 < 1 < …. < j < … < J


5 ordered logit model3

5. Ordered Logit Model

Ordered logit (see above)

  • 0 , J and 0: set equal to zero


5 ordered logit model4

5. Ordered Logit Model

Interpretation of parameters (marginal effects)


5 ordered logit model5

5. Ordered Logit Model

Estimation: ML


5 ordered logit model6

5. Ordered Logit Model

  • Disadvantages (Borooah 2002)

  • Assumption of equal slope k

  • Biased estimates if assumption of strictly ordered

    outcomes does not hold

  • treat outcomes as nonordered unless there are

    good reasons for imposing a ranking


5 ordered logit model7

5. Ordered Logit Model

Example: Effectiveness of better public transit as a way to reduce automobile congestion and air polution in urban areas

  • Research objective: develop and estimate models to measure how public transit affects automobile ownership and miles driven.

  • Data: Nationwide Personal Transportation Survey (42.033 hh): socio-demo’s, automobile ownership and use, public transportation avail.

Kim and Kim (2004)


5 ordered logit model8

5. Ordered Logit Model

  • Dependent variable ownership model = number of cars (k = 0, 1, 2,  3)  ordinal variable

  • C*i = latent variable: automobile ownership propensity of hh i

  • Relation to observed automobile ownership:

    Ci=k if k-1 < ’xi +  < k

  • P(Ci=k)=F(k- ’xi) - F(k-1 - ’xi)

Kim and Kim (2004)


Models with limited dependent variables

5. Ordered Logit Model


Models with limited dependent variables

5. Ordered Logit Model


Models with limited dependent variables

5. Ordered Logit Model


Models with limited dependent variables

5. Ordered Logit Model

  • Examples

  • Occupational outcome as a function of socio-demographic characteristics (Borooah)

    • Unskilled/semiskilled

    • Skilled manual/non-manual

    • Professional/managerial/technical

  • School performance (Sawkins 2002)

    • Grade 1 to 5

    • Function of school, teacher and student characteristics

  • Level of insurance coverage


D heterogeneity

D.Heterogeneity

  • Observed heterogeneity

  • Unobserved heterogeneity

    • Over decision makers

      • Random coefficients Models

      • E.g. Mixed Logit Model (see Train)

    • Over segments

      • Latent class estimation


D heterogeneity latent class est

D.Heterogeneity: Latent class est.

  • Ass.: Consumers can be placed into a small number of – homogeneous - segments which differ in choice behavior ( response parameters)

  • Relative size of the segment s (s=1, 2, ..., M) is given by

    fs = exp(s) / s’exp(s’)

Kamakura and Russell (1989)


D heterogeneity latent class est1

D.Heterogeneity: Latent class est.

  • Probability of choosing brand j, conditional on consumer i being a member of segment s

    Ps(yn = j|Xn) = exp(Xjns)

    lexp(Xlns)


D heterogeneity latent class est2

D.Heterogeneity: Latent class est.

  • Unconditional probability that consumer i will choose brand j

    P(yn = j|Xn) = sfs Ps(yn = j|Xn)

    = s exp(s) exp(Xjns)

    s’exp(s’) lexp(Xlns)


D heterogeneity latent class est3

D.Heterogeneity: Latent class est.

  • Estimation: Maximum Likelihood

  • Likelihood of a hh’s choice history Hn

    L(Hn) = s [ exp(s)L(Hn|s) / s’ exp(s’) ]

    with

    L(Hn|s) = t Ps(ynt = c(t) | Xnt)

    c(t) = index of the chosen option at time t.

  • Maximize likelihood over all hh’s: n L(Hn)


D heterogeneity latent class est4

D. Heterogeneity: Latent class est.

Segment analysis

  • Based on parameter estimates (e.g. difference in price sensitivity)

  • Based on segment profiles

    • Post-hoc: based on assignment of hh to segments; Probability that hh n belongs to segment s =

      P(ns | Hn) = L(Hn|s)fs / s’ [L(Hn|s’)fs’]

      Analyze characteristics of different segments

    • A priori: make fs a function of variables that may explain segment membership (e.g. income for segments which differ in price sensitivity)


D heterogeneity latent class est5

D.Heterogeneity: Latent class est.

  • Example: heterogeneity in price sensitivity (Bucklin and Gupta 1992)


D heterogeneity latent class est6

D.Heterogeneity: Latent class est.

  • Example: heterogeneity in price sensitivity


D heterogeneity latent class est7

D.Heterogeneity: Latent class est.


D heterogeneity latent class est8

D.Heterogeneity: Latent class est.

Choice segments: segment 1 = more sensitive to price and promo

Incidence segments: segment 2 and 4 = more sensitive to changes

in category attractiveness (change in price/promo)

 Confirms that  combinations of choice/inc.price sensitivity occur


D heterogeneity latent class est9

D.Heterogeneity: Latent class est.


D heterogeneity latent class est10

D. Heterogeneity: Latent class est.

  • Segment analysis: price elasticity


D heterogeneity latent class est11

D. Heterogeneity: Latent class est.

  • Segment analysis: socio-demographic profile


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